\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 112, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/112\hfil Multiplicity of solutions] {Multiplicity of solutions for some degenerate quasilinear elliptic equations} \author[V. Solferino \hfil EJDE-2009/112\hfilneg] {Viviana Solferino} \address{Viviana Solferino \newline Department of Mathematics, University of Calabria, Via P. Bucci, Arcavacata di Rende (CS), Italy} \email{solferino@mat.unical.it} \thanks{Submitted June 1, 2009. Published September 10, 2009.} \subjclass[2000]{35J20, 35J60, 35J70} \keywords{Quasilinear elliptic equations; variational methods; \hfill\break\indent noncoercive functionals; nonsmooth critical point theory} \begin{abstract} We show the existence of infinitely many solutions for a symmetric quasilinear problem whose principal part is degenerate. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition } \newtheorem{remark}[theorem]{Remark} \section{Introduction and statement of main result} Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with $n\geq 2$. We are interested in the solvability of the quasilinear elliptic problem $$\label{eq:q} \begin{gathered} -\mathop{\rm div}(j_\xi(x,u,\nabla u))+ j_s(x,u,\nabla u)=g(x,u) \quad \text{in } \Omega \,,\\ u=0\quad \text{on } \partial\Omega \,, \end{gathered}$$ where $j:\Omega \times\mathbb{R}\times \mathbb{R}^{n} \to \mathbb{R}$ is defined as $j(x,s,\xi)=\frac{1}{2}\,a(x,s)|\xi|^2\,.$ A feature of quasilinear problems like \eqref{eq:q} is that, for a general $u\in H^1_0(\Omega)$, the term $j_s(x,u,\nabla u)$ belongs to $L^1(\Omega)$ under reasonable assumptions, but not to $H^{-1}(\Omega)$. As a consequence, the functional $f(u) = \int_\Omega j(x,u,\nabla u)-\int_\Omega G(x,u)\,, \quad G(x,s)=\int_0^s g(x,t)\, dt\,,$ whose Euler-Lagrange equation is represented by \eqref{eq:q}, is continuous on $H^1_0(\Omega)$, but not locally Lipschitz, in particular not of class~$C^1$. In spite of this fact, existence and multiplicity results have been already obtained for this class of problems, also by means of variational methods, starting from the case in which $a$ is bounded and bounded away from zero (see \cite{Ar1, Ca}). Actually, in \cite{Ar1} the nonsmoothness of the functional is overcome by a suitable approximation procedure, while in \cite{Ca} a direct approach, based on a critical point theory for continuous functionals developed in \cite{ca-de,de-ma, ioffe_schwartzman1996, katriel1994}, is used. However, it seems to be hard, in the approach of \cite{Ar1}, to get multiplicity results when, e.g., $f$ is even. Both approaches have been extended to the case in which $a$ is still bounded away from zero, but possibly unbounded (unbounded case), in \cite{Ar2, Pe-Sq}, respectively. Again, multiplicity results when $f$ is even are proved only in the second paper. Finally, in \cite{ar-bo-or} the case in which $a$ is bounded, but not bounded away from zero (degenerate case) is addressed in the line of \cite{Ar1, Ar2}. Also in this case, no multiplicity result is proved when $f$ is even. The main purpose of this paper is to prove that, in the model case mentioned in \cite{ar-bo-or} and with a symmetry assumption, problem \eqref{eq:q} possesses infinitely many solutions. We also show that it is not necessary to restart all the machinery of the previous cases to get the result. By a change of variable, the degenerate case can be reduced to that of \cite{Ca}. Apart from the model case, it would be interesting to check if, up to a change of variable, the degenerate case can be reduced to a suitable form of the unbounded case and viceversa. To state our result, consider the model case $a(x,s) = \frac{1}{{(b(x)+s^2)}^\alpha}\,,$ where $b$ is a measurable function satisfying $0<\beta_1\leq b(x)\leq \beta_2$ a.e. in $\Omega$ and $\alpha \in [0,\frac{n}{2n-2})$. Let also $g:\Omega \times \mathbb{R}\to \mathbb{R}$ be a Carath\'{e}odory function satisfying the following assumptions: \begin{itemize} \item there exists $b,d >0$ and $22,R>0$ such that $$\label{cinque} 0<\nu G(x,s)\leq s g(x,s).$$ for almost every $x\in \Omega$ and all $s \in \mathbb{R}$ with $|s|\geq R$. It easily follows that the function $a$ satisfies the following conditions: \begin{itemize} \item there exist $c_1,c_2>0$ such that $$\label{uno} \frac{c_1}{{(1+|s|)}^{2\alpha}}\leq a(x,s)\leq c_2$$ for almost every $x \in \Omega$, for every $s \in \mathbb{R}$, \item for almost every $x$ in $\Omega$ the function $a(x,.)$ is differentiable on $\mathbb{R}$ and there exist $c_3>0$ such that, for almost every $x \in \Omega$, its derivative $$a_s(x,s)\equiv \frac{\partial a}{\partial s}(x,s)=\frac{-2\alpha\,s\,a(x,s)}{b(x)+s^2}$$ satisfies $$\label{due} |a_s(x,s)|\leq c_3 \quad \forall s \in \mathbb{R}$$ \item for almost every $x \in \Omega$ and all $s \in \mathbb{R}$ $$\label{quattro1} a(x,s)=a(x,-s).$$ \end{itemize} \begin{definition} \label{def1.1} \rm We say that $u$ is a weak solution of \eqref{eq:q} if $u \in H_0^1(\Omega)$ and $$\int_\Omega j_\xi(x,u,\nabla u)\nabla v+j_s(x,u,\nabla u)v =\int_\Omega g(x,u)\,v$$ for every $v \in C_0^\infty(\Omega)$. \end{definition} We are now able to state our main result. \begin{theorem}\label{undicio} Assume that conditions \eqref{quattro}, \eqref{alfa} and \eqref{cinque} hold. Then there exists a sequence $\{u_h\}\subset H_0^1(\Omega)\cap L^\infty(\Omega)$ of weak solutions of (\ref{eq:q}) such that $$\int_\Omega j(x,u_h,\nabla u_h)-\int_\Omega G(x,u_h)$$ approaches $+\infty$ as $h\to +\infty$. \end{theorem} \section{Proof of the main result} Let $\varphi \in C^2(\mathbb{R})$ be defined as $$\varphi(s)=s\,{(1+s^2)}^{\frac{\alpha}{2(1-\alpha)}}.$$ \begin{remark}\label{nota4} \rm We observe that $\varphi$ is odd and that there exists $\gamma>0$ such that $$\label{one} \varphi'(s)\geq \gamma(1+|\varphi(s)|)^\alpha.$$ Moreover we have $$\label{ai} \lim_{s\to \pm \infty} \frac{s\,\varphi'(s)}{\varphi(s)}= \lim_{s\to \pm \infty} \Big(1+s\frac{\varphi''(s)}{\varphi'(s)}\Big)=\frac{1}{1-\alpha}.$$ \end{remark} Let us consider the change of variable $u=\varphi(v)$. We can define on $H_0^1(\Omega)$ the functional $$\tilde{f}(v)=\frac{1}{2}\int_\Omega A(x,v)|\nabla v|^2- \int_\Omega \widetilde{G}(x,v),$$ where $$A(x,s)=a(x,\varphi(s))\cdot (\varphi'(s))^2, \quad \widetilde{G}(x,s)=G(x,\varphi(s))=\int_0^s \tilde{g}(x,t)dt,$$ with $\tilde{g}(x,s)=g(x,\varphi(s))\cdot \varphi'(s)$. Now let us consider the integrand $\tilde{j}: \Omega\times \mathbb{R}\times {\mathbb{R}}^n\to\mathbb{R}$ defined by $$\tilde{j}(x,s,\xi)=\frac{1}{2}A(x,s)|\xi|^2.$$ \begin{remark}\label{notaz2} \rm It is readily seen that \eqref{one} and the left inequality of (\ref{uno}) imply that for almost every $x \in \Omega$ and for every $(s,\xi)\in \mathbb{R}\times {\mathbb{R}}^n$, there holds $$\tilde{j}(x,s,\xi)\geq \alpha_0|\xi|^2,$$ where $\alpha_0=\frac{c_1 \,\gamma^2}{2}$. \end{remark} \begin{remark}\label{notazz} \rm By Remark \ref{nota4} there exists $\Lambda>0$ such that, for a.e. $x\in \Omega$ and for every $s \in \mathbb{R}$, we have \begin{gather}\label{del} A(x,s)\leq \Lambda, \\ \label{del1} |A_s(x,s)|\leq \Lambda. \end{gather} \end{remark} \begin{proposition}\label{mill1} Assume condition \eqref{cinque}. Then $$\label{cedo} \frac{\nu}{1-\alpha}\,\widetilde{G}(x,s)\leq s\tilde{g}(x,s)$$ for every $s\in \mathbb{R}$ with $|s|\geq R$. \end{proposition} \begin{proof} Condition (\ref{cinque}) implies $$\nu \widetilde{G}(x,s)\,\varphi'(s)\leq \varphi(s)\, \widetilde{G}_s(x,s)$$ hence $$\nu \frac{s \varphi'(s)}{\varphi(s)}\widetilde{G}(x,s)\leq s \widetilde{G}_s(x,s)$$ and taking into account Remark \ref{nota4}, we get the thesis. \end{proof} \begin{proposition}\label{notaz4} There exists $\mu< \frac{\nu}{1-\alpha}-2$ such that, for almost every $x \in \Omega$, for every $\xi \in {\mathbb{R}}^n$, for every $s \in \mathbb{R}$ with $|s|\geq R$, we have $$\label{cici} 0\leq s\tilde{j}_s(x,s,\xi)\leq \mu\, \tilde{j}(x,s,\xi).$$ \end{proposition} \begin{proof} Indeed \label{uu} \begin{aligned} \tilde{j}_s(x,s,\xi) &=\frac{1}{2}A_s(x,s)|\xi|^2\\ &= \frac{1}{2}[a_s(x,\varphi(s))\cdot (\varphi'(s))^3+2\varphi'(s)\cdot\varphi''(s)\cdot a(x,\varphi(s))]|\xi|^2\\ &=a(x,\varphi(s))\cdot \varphi'(s)\Big[\frac{-\alpha \,\varphi(s)\, (\varphi'(s))^2}{b(x)+(\varphi(s))^2}+\varphi''(s)\Big]|\xi|^2. \end{aligned} Let $s>0$. Then recalling that $a(x,\varphi(s))$ and $\varphi'(s)$ are positive functions, it suffices to prove that the square bracket is non negative. Note that the expression is equal to $$\frac{\alpha s{(1+s^2)}^{\frac{\alpha}{2(1-\alpha)}}}{(1-\alpha){(1+s^2)}^2} \Big[\frac{-(s^2+1-\alpha){(1+s^2)}^{\frac{\alpha}{(1-\alpha)}}}{b(x) +{(1+s^2)}^{\frac{\alpha}{(1-\alpha)}}\,s^2}+ \frac{(s^2+1-\alpha)\,b(x)}{(1-\alpha)(b(x)+{(1+s^2)} ^{\frac{\alpha}{1-\alpha}}\,s^2)}+2\Big].$$ Observing that the second term in square bracket is positive and the sum of the first and third is equal to $$\frac{{(1+s^2)}^{\frac{\alpha}{1-\alpha}}(s^2-(1-\alpha)) +2\,b(x)}{b(x)+{(1+s^2)}^{\frac{\alpha}{1-\alpha}}\,s^2},$$ the assertion follows if we assume $R=\sqrt{1-\alpha}$. On the other hand if $s<0$, taking into account that $\varphi''(s)$ is an odd function, we deduce that the square bracket in (\ref{uu}) is negative. Now we prove the right inequality. Since $\varphi'(s)\geq 0$ and $a_s(x,\varphi(s))\leq 0$, we have $$s\,\tilde{j}_s(x,s,\xi)\leq 2\,\tilde{j}(x,s,\xi)\Big(\frac{\varphi''(s)\,s}{\varphi'(s)}\Big)$$ and by Remark \ref{nota4} it follows the assertion with $R$ large enough and $\mu= \frac{2\,\alpha}{1-\alpha}$. \end{proof} We now are able to prove the main result of the paper. \begin{proof}[Proof of Theorem \ref{undicio}] By Remark \ref{notaz2} and \ref{notazz} and Proposition \ref{mill1} and \ref{notaz4} we are able to apply Theorem 2.6 in \cite{Ca}. So obtaining a sequence of weak solutions $\{v_h\} \subset H_0^1(\Omega)$ of the problem $$\int_\Omega A(x,v)\nabla v \nabla w+ \frac{1}{2}\int_\Omega A_s(x,v)|\nabla u|^2\,w= \int_\Omega \tilde{g}(x,v)w$$ for every $w \in C_0^\infty(\Omega)$ with $f(v_h)\to\infty$. By Theorem 7.1 in \cite{Pe-Sq} these solutions belong to $L^\infty(\Omega)$. If we set $u_h=\varphi(v_h)$, it is clear that $u_h\in H_0^1(\Omega)\cap L^\infty(\Omega)$ and an easy calculation shows that each $u_h$ is a weak solution of (\ref{eq:q}) with $$\int_\Omega j(x,u_h,\nabla u_h)-\int_\Omega G(x,u_h) \to +\infty$$ as $h\to +\infty$. \end{proof} \begin{thebibliography}{0} \bibitem{Ar1} D. Arcoya, L. Boccardo, \emph{Critical points for multiple integrals of the calculus of variations.} Arch. Rational Mech. Anal., \textbf{134}(1996),249--274. \bibitem{Ar2} D.Arcoya, L. Boccardo, \emph{Some remarks on critical point theory for nondifferentiable functionals}, NoDEA Nonlinear Differential Equations Appl. \textbf{6} (1999), 79-100. \bibitem{ar-bo-or} D. Arcoya, L. Boccardo, L. Orsina, \emph{Existence of critical points for some noncoercive functionals}, Ann. I. H. Poincar\'{e} Anal. Nonlin\'{e}aire., \textbf{18},no.4 (2001), 437-457. \bibitem{Ca} A. 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